Block ciphers should be designed to resist all classical attacks. S-boxes are the core components of block ciphers. The primary purpose of S-boxes is to produce confusion inside block ciphers.
Such S-boxes are nonlinear functions over \(\mathbb {F}_{2^{n}}\) . These functions should have low differential uniformity for resisting differential attacks [1]}.
In cryptography, APN functions over \(\mathbb {F}_{2^{n}}\)  are the optimal ones to resist differential attacks.
Therefore, APN functions have important applications in block ciphers.
For example, the APN functions \(x^{81}\)  over \(\mathbb {F}_{2^9}\)  and \(x^5\)  over \(\mathbb {F}_{2^7}\)  have been respectively used in MISTY and KASUMI block ciphers. APN power functions over finite fields attract researchers' interest for their simple algebraic form and some additional properties. So far, researchers only found six classes of APN power functions on \(\mathbb {F}_{2^n}\) : Gold functions [2]}, Kasami functions [3]}, Welch functions [4]}, Niho functions [5]}, Inverse functions [6]} and Dobbertin functions [7]}. Furthermore, many results of APN functions appeared in the recent years, the reader may refer to [8]} and references therein for more information.
